Properties

Label 2-1452-1.1-c3-0-50
Degree $2$
Conductor $1452$
Sign $-1$
Analytic cond. $85.6707$
Root an. cond. $9.25585$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 21.4·5-s − 28.4·7-s + 9·9-s − 53.5·13-s + 64.2·15-s + 44.9·17-s − 118.·19-s − 85.2·21-s − 148.·23-s + 334.·25-s + 27·27-s + 124.·29-s − 24.8·31-s − 608.·35-s − 20.0·37-s − 160.·39-s − 35.8·41-s − 290.·43-s + 192.·45-s − 216.·47-s + 464.·49-s + 134.·51-s − 656.·53-s − 355.·57-s − 204.·59-s − 8.56·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.91·5-s − 1.53·7-s + 0.333·9-s − 1.14·13-s + 1.10·15-s + 0.641·17-s − 1.43·19-s − 0.885·21-s − 1.34·23-s + 2.67·25-s + 0.192·27-s + 0.798·29-s − 0.143·31-s − 2.94·35-s − 0.0891·37-s − 0.659·39-s − 0.136·41-s − 1.03·43-s + 0.638·45-s − 0.671·47-s + 1.35·49-s + 0.370·51-s − 1.70·53-s − 0.826·57-s − 0.451·59-s − 0.0179·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(85.6707\)
Root analytic conductor: \(9.25585\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1452,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - 3T \)
11 \( 1 \)
good5 \( 1 - 21.4T + 125T^{2} \)
7 \( 1 + 28.4T + 343T^{2} \)
13 \( 1 + 53.5T + 2.19e3T^{2} \)
17 \( 1 - 44.9T + 4.91e3T^{2} \)
19 \( 1 + 118.T + 6.85e3T^{2} \)
23 \( 1 + 148.T + 1.21e4T^{2} \)
29 \( 1 - 124.T + 2.43e4T^{2} \)
31 \( 1 + 24.8T + 2.97e4T^{2} \)
37 \( 1 + 20.0T + 5.06e4T^{2} \)
41 \( 1 + 35.8T + 6.89e4T^{2} \)
43 \( 1 + 290.T + 7.95e4T^{2} \)
47 \( 1 + 216.T + 1.03e5T^{2} \)
53 \( 1 + 656.T + 1.48e5T^{2} \)
59 \( 1 + 204.T + 2.05e5T^{2} \)
61 \( 1 + 8.56T + 2.26e5T^{2} \)
67 \( 1 - 134.T + 3.00e5T^{2} \)
71 \( 1 + 260.T + 3.57e5T^{2} \)
73 \( 1 + 528.T + 3.89e5T^{2} \)
79 \( 1 - 834.T + 4.93e5T^{2} \)
83 \( 1 + 617.T + 5.71e5T^{2} \)
89 \( 1 + 403.T + 7.04e5T^{2} \)
97 \( 1 + 1.18e3T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.013162367773425239950733313469, −8.075940392009883339933490400714, −6.80661339724241178584526255572, −6.38698072573243245540279980235, −5.61904866398394710137548694594, −4.55867661085310352955329580268, −3.22846694948864126415516078177, −2.49861834453438752216727843561, −1.68095426589459545084822919458, 0, 1.68095426589459545084822919458, 2.49861834453438752216727843561, 3.22846694948864126415516078177, 4.55867661085310352955329580268, 5.61904866398394710137548694594, 6.38698072573243245540279980235, 6.80661339724241178584526255572, 8.075940392009883339933490400714, 9.013162367773425239950733313469

Graph of the $Z$-function along the critical line